Mathematics > Algebraic Geometry

Title:Flops and Clusters in the Homological Minimal Model Program

Abstract: Suppose that $f\colon X\to\mathrm{Spec}\, R$ is a minimal model of a complete
local Gorenstein 3-fold, where the fibres of $f$ are at most one dimensional,
so by [VdB1d] there is a noncommutative ring $Λ$ derived equivalent to
$X$. For any collection of curves above the origin, we show that this
collection contracts to a point without contracting a divisor if and only if a
certain factor of $Λ$ is finite dimensional, improving a result of [DW2].
We further show that the mutation functor of [S6][IW4] is functorially
isomorphic to the inverse of the Bridgeland--Chen flop functor in the case when
the factor of $Λ$ is finite dimensional. These results then allow us to
jump between all the minimal models of $\mathrm{Spec}\, R$ in an algorithmic
way, without having to compute the geometry at each stage. We call this process
the Homological MMP.
This has several applications in GIT approaches to derived categories, and
also to birational geometry. First, using mutation we are able to compute the
full GIT chamber structure by passing to surfaces. We say precisely which
chambers give the distinct minimal models, and also say which walls give flops
and which do not, enabling us to prove the Craw--Ishii conjecture in this
setting. Second, we are able to precisely count the number of minimal models,
and also give bounds for both the maximum and the minimum numbers of minimal
models based only on the dual graph enriched with scheme theoretic
multiplicity. Third, we prove a bijective correspondence between maximal
modifying $R$-module generators and minimal models, and for each such pair in
this correspondence give a further correspondence linking the endomorphism ring
and the geometry. This lifts the Auslander--McKay correspondence to dimension
three.

Comments:

58 pages. Last update had an old version of Section 4, no other changes. Final version, to appear Invent. Math